Is there a way to make a button disappear on a page after it's been clicked on?

if you realy want to get rid of that button – you can use remove for it – place that in child-theme functions.php
f.e. you placed an enfold button and give a custom class to it f.e.: button-customclass

function remove_a_button_with_custom_class(){
?>
<script>
(function($){	
	$(".button-customclass a").click(function(){
    	$(this).remove();
	});
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'remove_a_button_with_custom_class');